1. Field of the Invention
The present invention relates to a method of upscaling absolute permeability values associated with a fine scale model representative of a porous medium to the inter-cell absolute permeability values associated with a model representative of the porous medium at the flow simulation scale.
2. Description of the Prior Art
The objective of reservoir engineering is to predict the dynamic behavior of the subsoil in order to maximize hydrocarbon production. Models representative of the reservoirs, as accurate as possible in relation to the data collected (logs, seismic data, outcrop data, production data, . . . ), therefore have to be constructed and the production of these constrained reservoirs has to be estimated by flow simulations. It is desired, insofar as a reservoir model compatible with the whole of the data is available, to reduce the uncertainty on the production predictions. Prior to flow simulation, the reservoir is characterized in terms of geometry and petrophysical properties in form of numerical models. In the case of complex models containing a large number of details, the numerical simulations may require considerable computer resources. They have to be reduced for economic reasons while maintaining a good compromise between quickness and result accuracy.
When a physical phenomenon has to be studied, the physical equations assumed to be representative of this phenomenon and a N-dimensional domain (N>0 is generally two or three) on which the phenomenon in question acts first have to be determined. The physical phenomena encountered are of very diverse natures, examples thereof being: reaction, convective or diffusive mechanisms.
The equations are generally ordinary differential equations or partial differential equations. The system of equations thus defined is commonly referred to as physical model of the phenomenon. Numerical simulation of this model solves numerically the system of equations for the model in question. The equations of the model are conventionally solved using finite volume, finite element and mixed finite element type approaches. Once the equation solution approach and the domain are selected, the system of equations is discretized according to a sampling defining points over the entire domain. These points are also referred to as grid nodes or simply nodes. A link acts as a bond between two neighboring nodes. A cell is defined by the volume contained in a closed surface connecting several grid nodes. A face is the link between two neighboring cells. The grid of a domain contains all of the nodes in the domain. For each physical parameter, a value thereof is associated with either a cell or a node, depending on the equation discretization approach which is used. According to the nature of the physical phenomenon studied, certain equation solution approaches are more suitable than others. For example, when a pure diffusive problem is to be solved, finite elements provide more accurate results than finite volume type approaches. On the other hand, finite volume approaches allow for better accounting of convective and diffusive phenomena.
In petroleum engineering, most numerical models are associated with a structured grid. Within a structured grid, a cell has a fixed number of faces and thus of neighboring cells. This type of grid is simple to construct and writing of the equations is relatively simple. Examples thereof are Cartesian grids, radial grids or Corner Point Geometry type grids. Such grids are for example described in the following documents:                Pointing, D. K., 1989, Corner Point Geometry in Reservoir Simulation, Joint IMA/SPE European Conference on the Mathematics of Oil Recovery,        Balaven-Clermidy, S., 2001, “Génération de Maillages Hybrides Pour la Simulation des Réservoirs Pétroliers”, PhD Thesis, Ecole des Mines, Paris.        
However, these grids do not offer great flexibility, notably to account for the faults, the wells and the spatial variability of the physical properties. To overcome this problem, reservoir engineers use unstructured grids. In an unstructured grid, the cells have a variable number of faces, therefore of neighboring cells. Such grids are known in the art. Examples thereof are Voronoi type grids and grids with local refinements. Such grids are for example described in the following documents:                Palagi, C. L. and Aziz, K.: “Use of Voronoi Grid in Reservoir Simulation”, paper SPE 22889 presented at the 66th Annual Technical Conference and Exhibition, Dallas, Tex., 1991,        Lepage, F.: “Génération de Maillages Tridimensionnels Pour la Simulation des Phénomènes Physiques en Géoscience”, PhD Thesis, Institut National Polytechnique de Lorraine, 2003.        
In this case, the equation solution methods are more complex to write, but the geometry of the complex objects of the domain is better reproduced than with structured grids. The nature of the physical phenomena involved in petroleum engineering leads engineers to preferably use, for flow simulations, approaches based on the calculation of flows between the cells. Examples of the approaches used are the mixed finite elements and the finite volumes. Such techniques are for example described in the following documents:                Brezzi, F. and Fortin, M., “Mixed and Hybrid Finite Element Method”, Springer-Verlag, New York, 1991,        Eymard, R., Gallouët, T. And Herbin, R., “Finite Volume Methods”, in Ciarlet, P. G. and Lions, J. L., “Handbook of Numerical Analysis”, 2000.        
These approaches require calculation of the flows of conservative quantities through each face of the grid cells. Values therefore have to be available for the petrophysical properties at the faces. The petrophysical properties that can be studied are, for example, the porosity, the absolute permeability, the relative oil, oil and gas permeabilities, the fluid velocities, the capillary pressure or the saturation of the various fluids.
Construction of a numerical model of the subsoil compatible with all the data available can lead to a numerical model comprising a very large number of details. This numerical model is commonly referred to as geologic model. The stage of flow simulation on such a model is then very slow because the computing time required is expressed, at best, as a linear function of the number of cells. It is then commonplace in oil reservoir engineering to resort to the creation of a new numerical model whose associated grid comprises less cells. This new numerical model will be referred to as reservoir model hereafter. The stage of flow simulation on the reservoir model requires acceptable computing times. During construction of the reservoir model, it is necessary to assign the associated petrophysical property values to each cell of the grid of this model. For a numerical model, a geometrical quantity characteristic of each cell associated with the model is defined. This quantity is called scale. In a numerical model, a property is known at a given scale. When using the reservoir model, the scale of the properties of the medium changes. A stage of upscaling the properties in question is then necessary.
Since the dynamic behavior of the subsoil is very complex, during the upscaling procedure, successive stages are used to calculate one after the other the various petrophysical properties that will make up the reservoir model. The first one is the absolute permeability. The absolute permeability is the petrophysical property that accounts for the ability of the rock to allow the fluid to flow through the pores thereof. Considering the complex geologic structure of the subsoil, the absolute permeability is a spatially heterogeneous property. It is generally calculated, for the geologic model, by means of samples taken in wells. The value of the absolute permeability associated with each cell of the geologic model is assumed to be constant on the cell in question. Since the absolute permeability is a non-additive variable, upscaling thereof cannot be considered from a simple law. This problem has led to the development of many techniques such as heuristic approaches, the homogenization theory or numerical approaches. These techniques are for example described in the following documents:                Renard, Ph. And Marsily, G. de, 1997, Calculating Equivalent Permeability: A Review. Advances in Water Resources, 20:253-278,        Durlofsky, L. J., 2003, Upscaling of Geocellular Models for Reservoir Flow Simulation: A Review of Recent Progress, Baden-Baden, Germany, 7th International Forum on Reservoir Simulation.        
Homogenization methods were developed for analysis of statistically homogeneous permeability fields and heuristic methods for particular configurations such as stratified media or lognormal distribution two-dimensional media. In any case, resorting to numerical methods is imperative. These methods are based on the definition of one or more equivalence criteria. The criteria generally used do not directly apply to permeabilities, but to additive and conservative variables such as the mass or the energy dissipated by the system. Computation of these criteria requires fine-scale and sometimes coarse-scale single-phase flow simulations. In fact, these upscaling methods are referred to as numerical.
Numerical methods for absolute permeability upscaling can be classified according to the size of the domain of the geologic model involved in the flow simulations. To carry out a flow simulation, the absolute permeability values and the associated grid contained in the domain in question are extracted from the geologic model. It is referred to as a local method when the domain is equal to the volume of the cell of the reservoir model whose associated permeability value is to be calculated. Neighborhood-based methods involve a domain that is larger than the volume of the cell of the reservoir model whose associated permeability value is to be calculated. Finally, for global methods, the domain of the flow simulation is the complete geologic model. Unlike local methods and neighborhood-based methods, global methods allow to apprehend the large-scale connectivity that is observed in the geologic model. Such techniques are described for example in the following documents:                Khan, S. A. and Dawson, A. G., Methods of Upscaling Permeability for Unstructured Grids, U.S. Pat. No. 6,826,520 B1, 30 Nov. 2004,        Pickup, G. E., Jensen, P. S., Ringrose, P. S. and Sorbie, K. S., A method for Calculating Permeability Tensors Using Perturbed Boundary Conditions, Third European Conference on the Mathematics of the Oil Recovery, 1992,        Chen, Y., Durlofsky, L. J., Gerritsen, M. And Wen X. H., A Coupled Local-Global Upscaling Approach for Simulating Flow in Highly Heterogeneous Formations, Advances in Water Resources, 26(2003) 1041-1060.        
Flow simulations on the reservoir model involve approaches based on the calculation of flows between the cells. In order to evaluate these flows, the petrophysical properties along the faces of the cells have to be known. The inter-cell transmissivities are commonly used to calculate the flows. The inter-cell transmissivity allows quantification of the numerical flow passing through the face in question. It depends on the flow simulation solution method and on the value of the inter-cell absolute permeability. The inter-cell absolute permeability is defined as the value of the absolute permeability along the face. Calculation of the inter-cell absolute permeability values from the absolute permeability values of the cells cannot be considered from a simple law within a general context. Approximate formulas exist. Such techniques are for example described in the following documents:                Peaceman, D. W., 1977, Fundamentals of Numerical Reservoir Simulation, Elsevier scientific pub., New York,        Journel, A. G., Deutsch, C. And Desbarats, A. J., 1986, Power Averaging for Block Effective Permeability, paper SPE15128 presented at the 56th California Regional Meeting of SPE, Society of Petroleum Engineers, Long Beach.        
However, an error on the estimation of the inter-cell permeability is made. This result is underscored in the following document:                Romeu, R. K. and Noetinger, B., 1995, Calculation of Internodal Transmissivities in Finite Difference Models of Flows in heterogeneous Porous Media, Water Resources Research, 31(4), 943-959.        
An important point of the upscaling stage is the nature of its result. Existing methods of absolute permeability upscaling calculate either the absolute permeability values associated with the grid cells of the reservoir model, or the inter-cell transmissivity values. There is no method for determining the inter-cell permeability values of the reservoir model.
The method allowing upscaling of a geologic model, that is to construct a numerical reservoir model representative of a porous heterogeneous medium, therefore has to meet several requirements:                1. The method must be independent of the structure and of the shape of the grid associated with the numerical reservoir model.        2. The method must allow, if need be, to reproduce on the reservoir model the large-scale connectivity of the geologic model.        3. The method must limit flow simulation errors through the use of the inter-cell permeabilities.        